Under the assumption that this assertion is true (Conjecture 3.3), we say that (Sn) satisfies the Restricted Prime Number Theorem

AMONG THE SUMS OF THE FIRST 2n PRIMES

ROMEO ME ˇSTROVI ´C

ABSTRACT. Let p_nbe nth prime, and let (S_n)^∞_n=1:= (S_n) be the sequence of the sums of the first 2n consecutive primes, that is, Sn=P2n

k=1pkwith n = 1, 2, . . .. Heuristic arguments supported by the corresponding computational results suggest that the primes are distributed among sequence (Sn) in the same way that they are distributed among positive integers. In other words, taking into account the Prime Number Theorem, this assertion is equivalent to

#{p : p is a prime and p = Sk for some k with 1 ≤ k ≤ n}

∼#{p : p is a prime and p = k for some k with 1 ≤ k ≤ n} ∼ log n

n as n → ∞, where |S| denotes the cardinality of a set S. Under the assumption that this assertion is true (Conjecture 3.3), we say that (Sn) satisfies the Restricted Prime Number Theorem.

Motivated by this, in Sections 1 and 2 we give some definitions, results and examples concerning the generalization of the prime counting function π(x) to increasing positive integer sequences.

The remainder of the paper (Sections 3-7) is devoted to the study of mentioned se- quence (Sn). Namely, we propose several conjectures and we prove their consequences concerning the distribution of primes in the sequence (Sn). These conjectures are mainly motivated by the Prime Number Theorem, some heuristic arguments and related computational results. Several consequences of these conjectures are also established.

1. INTRODUCTION, MOTIVATION ANDPRELIMINARIES

An extremely difficult problem in number theory is the distribution of the primes among the natural numbers. This problem involves the study of the asymptotic behavior of the counting function π(x) which is one of the more intriguing functions in number theory. The function π(x) is defined as the number of primes ≤ x. For elementary methods in the study of the distribution of prime numbers, see [12].

Although questions in number theory were not always mathematically en vogue, by the middle of the nineteenth century the problem of counting primes had attracted the at- tention of well-respected mathematicians such as Legendre, Tch´ebychev, and the prodi- gious Gauss.

A query about the frequency with which primes occur elicited the following response:

I pondered this problem as a boy, and determined that, at aroundx, the primes occur with density1/ log x–C. F. Gauss (letter to Encke, 24 December 1849). Gauss wrote:

This remark of Gauss can be interpreted as predicting that

#{primes ≤ x} ≈

bxc

X

n=2

1 log n ≈

Z x 2

dt

log t = Li(x).

2010 Mathematics Subject Classification. Primary 11A41, Secondary 11A51, 11A25.

Keywords and phrases: distribution of primes, prime counting function, Prime Number Theorem, prime-like sequence, ω-Restricted Prime Number Theorem.

1

Studying tables of primes, C. F. Gauss in the late 1700s and A.-M. Legendre in the early 1800s conjectured the celebrated Prime Number Theorem:

π(x) = |{p ≤ x : p prime}| ∼ x log x (here, as always in the sequel, |S| denotes the cardinality of a set S).

This theorem was proved much later ([8, p. 10, Theorem 1.1.4]; for its simple analytic proof see [31] and [46], and for its history see [3], [21], [22] and [28, p. 21]. Briefly, π(x) ∼ x/ log x as x → ∞, or in other words, the density of primes p ≤ x is 1/ log x;

that is, the ratio π(x) : (x/ log x) converges to 1 as x grows without bound. Using L’Hˆopital’s rule, Gauss showed that the logarithmic integral Rx

2 dt/ log t, denoted by Li(x), is asymptotically equivalent to x/ log x. Recall that Gauss felt that Li(x) gave better approximations to π(x) than x/ log x for large values of x.

Though unable to prove the Prime Number Theorem, several significant contribu- tions to the proof of Prime Number Theorem were given by P. L. Chebyshev in his two important 1851–1852 papers ([6] and [7]). Chebyshev proved that there exist positive constants c1 and c2 and a real number x0 such that c1x/ log x ≤ π(x) ≤ c1x/ log x for x > x0. In other words, π(x) increases as x log x. Using methods of complex analysis and the ingenious ideas of Riemann (forty years prior), this theorem was first proved in 1896, independently by J. Hadamard and C. de la Vall´ee-Poussin (see e.g., [33, Section 4.1]).

A generalized prime system (or g-prime system) G is a sequence of positive real num- bers q₁, q₂, q₃, . . . satisfying 1 < q₁ ≤ q₂ ≤ · · · ≤ q_n ≤ q_n+1 ≤ · · · and q_n → ∞ as n → ∞. From these can be formed the system N of generalized integers or Beurling integers; that is, the numbers of the form q^k₁¹q^k₂^l· · · q^k_l^l, where l ∈ _Nand k₁, k₂, . . . , k_l ∈

N0 :=NS{0}. Notice that N denotes the multiplicative semigroup generated by G, and it consists of the unit 1 together with all finite power-products of g-primes, arranged in increasing order and counted with multiplicity.

Clearly, this system generalizes the notion of primes and positive integers obtained from them. Such systems were first introduced by A. Beurling [5] and have been studied by many authors since then (see in particular [4], [2], [11], [13], [25], [32] and [47]). In particular, Nyman [32] and Malliavin [25] sharpened Beurling’s results in various ways.

Much of the theory concerns connecting the asymptotic behaviour of g-prime counting functionand g- counting function πG(x) and NG(x), defined on [1, ∞) respectively by

πG(x) = X

q∈G,q≤x

1 and NG(x) = X

n∈N ,n≤x

1,

where in the first sum the summation is taken over all g-primes, counting multiplicities.

Similarly, for the second sumP

n∈N ,n≤x1. Accordingly, we have

πG(x) = #{i : q_i ∈ G and g_i ≤ x} and NG(x) = #{i : n_i ∈ N and n_i ≤ x}.

If G = {a₁, a₂, . . . , a_n, a_n+1, . . .} = (a_n)^∞_n=1 is a sequence such that a₁ ≤ a₂ ≤ · · · ≤ a_n ≤ a_n+1 ≤ · · · , then obviously, we have πG(a_n) = n for each n ∈N.

In 1937 Beurling proved [5, Th´eor`eme IV] that if NG satisfies the asymptotic relation NG(x) = Ax + O(x/ log^γx) with some constants A > 0 and γ > 3/2, then the number of qn’s such that qn≤ x is equal to x/ log x + o(x/ log x), i.e.,

πG(x) = x log x+ o

 x

log x

 .

In other words, the conclusion of the Prime Number Theorem (in the sequel, shortly writ- ten as PNT) is valid for such a system G (from this reason often called a Beurling prime number system). Beurling also gave an example in which NG(x) = Ax + O(x/ log^3/2x) but PNT is not valid. This result was refined in 1969 by H. G. Diamond [9, Theorem (B)]. In 1970 Diamond [10] also proved that Beurling’s condition is sharp, namely, the PNT does not necessarily hold if γ = 3/2.

In particular, if G is a set P := {p₁, p₂, p₃, . . .} of all primes 2 = p₁ < p₂ < p₃ < · · · with the associated multiplicative semigroup N = N = {1, 2, 3, . . .}, then PNT states that

π(x) ∼ x

log x, as x → ∞, where π(x) is the usual prime counting function, that is,

π(x) = X

p prime, p<x

1.

As observed in [4, Introduction], the additive structure of the positive integers is not particularly relevant to the distribution of primes. Therefore, for a given g-prime sys- tem G defined above, it can be of interest to consider the distribution of g-primes (the elements in G) with respect to certain associated system of generalized integers without any algebraic (multiplicative) structure. This means that the associated system N to G defined above may be some subset of [1, +∞) which is not a multiplicative semigroup (generated by G).

In particular, here we mainly consider the case when G is an infinite set of primes and the associated system N to G is an increasing integer sequence (a_n)^∞_n=1. We focus our attention when G is a set of all primes whose associated system N is the sequence (a_n)^∞_n=1:= (P2n

i=1p_i)^∞_n=1where 2 = p₁ < p₂ < · · · < p_n< · · · are all the primes.

Let (G, N := (a_k)^∞_k=1) be a pair defined above. Then we define its counting function NG,(a_k)(x) as

NG,(a_k)(x) = #{i : i ∈Nand a_i ≤ x}.

Furthermore, the prime counting function for (G, N := (a_k)^∞_k=1) is the function x 7→

πG,(a_k)(x) defined on [1, ∞) as

(1) πG,(a_k)(x) = #{q : q ∈ G and q = a_i for some i with a_i ≤ x}.

Heuristic and computational results show that for many “natural pairs” (G, N := (a_k)^∞_k=1) the associated counting function NG,(a_k)(x) has certain asymptotic growth as x → ∞.

Notice that for each n ∈Nwe have

(2) πG,(a_k)(a_n) = #{q : q ∈ G and q = a_i for some i with 1 ≤ i ≤ n}.

The normalizable prime counting function for (G, N = (ak)^∞_k=1) is the function (n, x) 7→

p_G,(ak)(n, x) defined for (n, x) ∈_N× [1, +∞) as (3) pG,(a_k)(n, x) = log a_n

a_n πG,(a_k)(x).

The above expression induces the companion sequence (bn)^∞_n=1of (ak)^∞_k=1defined as (4) b_n= pG,(a_k)(a_n) = log an

a_n πG,(a_k)(a_n), n = 1, 2, . . . .

We also define another “companion” sequence (c_n)^∞_n=1of (a_k)^∞_k=1 defined as

(5) cn = a_n

(log n)(log a_n) = πG,(a_k)(a_n)

b_nlog n , n = 1, 2, . . . .

Here as always in the sequel, we will suppose that G is a set of all primes whose associated system N is a sequence (a_k)^∞_k=1. For brevity, in the sequel the functions defined by (1), (2), (3) and (4) will be denoted by p_(ak)(x), π_(ak)(a_n), p_(ak)(n, x) and p_(ak)(a_n), respectively.

Definition 1.1. Let Ω be a set of all nonnegative continuous real functions defined on (1, +∞) and let (a_k)^∞_k=1 := (a_k) be an increasing sequence of positive integers. We say that (ak) is a prime-like sequence if there exists the function ω_(ak) = ω ∈ Ω such that the function n 7→ π_(ak)(a_n) defined by (2) is asymptotically equivalent to ω(n) as n → ∞.

Then we say that a sequence (a_k) satisfies the ω-Restricted Prime Number Theorem.

In particular, if ω(x) ∼ x/ log x as x → ∞, then we say that a sequence (a_k) satisfies the Restricted Prime Number Theorem (RPNT).

Proposition 1.2. Let (a_k)^∞_k=1 be a positive integer sequence, and let(b_n)^∞_n=1be its com- panion sequence defined by(4). Then

(6) lim sup

n→∞

b_n≤ 1.

Proof. Taking the obvious inequality π_(ak)(a_n) ≤ π(a_n) with n = 1, 2, . . . into (4) we get

b_n≤ π(a_n) log a_n

a_n , n = 1, 2, . . . , which by the Prime Number Theorem immediately yields

lim sup

n→∞

b_n ≤ lim

n→∞

π(an) log an

a_n = 1,

as desired. 

Proposition 1.3. Let (a_n) be a prime-like sequence with the associated function ω(x).

Then

(7) lim sup

n→∞

ω(n) π(a_n) ≤ 1.

This means thatω(n) grows slowly than π(a_n) as n → ∞.

Proof. Notice that the inequality (7) is equivalent to

(8) lim sup

n→∞

log an

a_n π_(ak)(a_n) ≤ 1.

Since by the assumption, ω(n) ∼ π_(ak)(a_n), the inequality (8) yields lim sup

n→∞

log a_n

a_n ω(n) ≤ 1,

whence, in view of the fact that log a_n/a_n∼ 1/π(a_n), immediately follows (7).  Remark 1.4. The inequality (6) is sharp since by the Prime Number Theorem (see Ex- ample 2.1), equality in (6) holds for the sequences ak = k with k = 1, 2, . . ..

Remark1.5. If a sequence (a_k) satisfies the ω-Restricted Prime Number Theorem, then by (4) we have

(9) ω(n) ∼ π_(ak)(an) = anbn

log a_n as n → ∞.

Remark1.6. Let (ak) be a sequence satisfying the ω- Restricted Prime Number Theorem.

Then clearly, ω(a_k)(n) ≤ n for all sufficiently large n. Moreover, ω(a_k)(n) ∼ n as n → ∞ if and only if the density of primes in a sequence (a_k) is equal to 1.

Notice also that by the Prime Number Theorem, ω_N(x) = x/ log x for the sequence of all positive integers N = {1, 2, . . . n, . . .}, that is, _N satisfies the Restricted Prime Number Theorem (cf. Conjecture 3.3).

Here, as always in the sequel, P = (pn) := {p₁, p₂, . . . p_n, . . .} will denote the set of all primes, where 2 = p1 < 3 = p₂ < p₃ < · · · < p_n< · · · . Moreover, (a_n) will always denotes an infinite strictly increasing sequence of positive integers. Hence, for such a sequence must be a_n ≥ n for each n ∈_N.

The remainder of the paper is organized as follows. In Section 2 we present five examples concerning the determination of the function ω_(ak)(x) and a sequence (b_n) associated to a given sequence (a_k). In particular, we consider the sequence (a_k)^∞_k=1 with a_k= a + (k − 1)d, where a ≥ 1 and d > 1 are relatively prime integers.

In Section 3 we consider the distribution of primes in the sequence (S_n)^∞_n=1 whose terms are given by S_n = P2n

i=1p_i, where p_i is the ith prime. Heuristic arguments sup- ported by related computational results suggest the curious conjecture that the sequence (S_n) satisfies the Restricted Prime Number Theorem (Conjecture 3.3). In other words, this means that the primes are distributed amongst all the terms of the sequence (S_n) in the same way that they are distributed amongst all the positive integers. Under this con- jecture, we prove that if q_kis the kth prime in (S_n)^∞_n=1, then q_k ∼ 2k²log³k ∼ 2p²_klog k as k → ∞ (Corollaries 3.6 and 3.7).

Assuming that Conjecture 3.3 is true, in Section 4 we give the asymptotic expression for the kth prime in the sequence (Sn) (Corollary 4.2); namely, qk ∼ 2k²log³k as k → ∞. This result is refined by Theorem 4.4. We also conjecture that bk log kc+1 ≤ m for each pair (k, m) of positive integers with k ≥ 1 and qk= S_m(Conjecture 4.6). Some consequences of Conjectures 3.3 and 4.6 are also presented.

Section 5 is devoted to the estimations of the values Mk (k = 1, 2, . . .) involving in the expression for q_k from Theorem 4.4. We also propose some other conjectures concerning the sequences (S_k) and (M_k). Related consequences are also established.

The conjectures presented in this paper, as well as some their consequences, are mainly supported by some computational results given in Section 6. In particular, the number π_n := k of primes in the set S_n := {S₁, S₂, . . . , S_n} for 38 values of n up to 10⁹+ 5 · 10⁸ are presented in Table 1. For such values k and the associated indices m such that q_k= S_m, the corresponding approximate values of q_k, M_k(together with lower and upper bounds of Mk), (k log k)/m and Sm

√k log k/(2m^5/2log m) are also given in this table. Under the previous notations, related numerical results for q_k/(2k²log³k), q_k/(2m²log m) and two estimates involving q_k which are discussed in Section 4, are given in Table 3. Some additional computational results, the conjectures and their con- sequences are also given in Section 6.

In the last Section 7 we propose the stroner (asymptotic) version of Conjecture 3.3 which coincides with well known form of Prime Number Theorem involving the function li(x).

Notice that similar considerations to those for the sequence (S_n) concerning alternat- ing sums of consecutive primes are given in [29].

2. EXAMPLES

Example 2.1. For the sequence (a_k) with a_k = k (k = 1, 2, . . .), we clearly have π_(k)(n) = π(n), and hence

(10) b_n= π(n) log n

n , n = 1, 2, . . . . By the Prime Number Theorem, from (10) we find that

(11) lim

n→∞bn = lim

n→∞

π(n) log n

n = 1.

Example 2.2. Let (a_k) be a sequence of all primes, that is, a_k = p_k with k ∈ _N :=

{1, 2, . . .}, where p_k is the kth prime. Since by (1), π_(pk)(p_n) = π(p_n) = n, substituting this into (4) yields

(12) b_n= n log p_n

p_n , n = 1, 2, . . . .

Now applying to (12) the well known fact that pn ∼ n log n as n → ∞ (see, e.g., [30]), we find that

(13) lim

n→∞b_n= 1.

Notice also that the known inequality p_n > n log n with n ≥ 1 (see, e.g., [38, (3.10) in Theorem 3]) implies that b_n< 1 for all n ≥ 1.

Example 2.3. Suppose that a and d are relatively prime positive integers. Then concern- ing Dirichlet’s theorem de la Vall´ee Poussin established (see, e.g., [35, p. 205]) that the number of primes p < x with p ≡ a(mod d) is approximately

(14) π(x)

ϕ(d) ∼ 1

ϕ(d) · x log x.

Here ϕ(n) is the Euler totient function defined as the number of positive integers not exceeding n and relatively prime to n. Note that the right hand side of (14) is the same for any a such that gcd(a, d) = 1. This shows that primes are in a certain sense uniformly distributed in reduced residue classes with respect to a fixed modulus. Notice that for a sequence (a_k)^∞_k=1given by a_k= a + (k − 1)d, (14) can be written as

(15) π_(ak)(a_n) ∼ π(a_n)

ϕ(d) as n → ∞.

Inserting (15) together with π(a_k) ∼ a_k/ log a_kinto (4) immediately gives

(16) lim

n→∞b_n= 1 ϕ(d) lim

n→∞

π(a_n) log a_n

a_n = 1

ϕ(d).

Then substituting (16) into (9), we obtain that for the associated function ω_a,d(x) := ω_(ak of the sequence (a_k) there holds

(17)

ω_a,d(n) ∼ π_(ak)(a_n) a_nb_n

ϕ(d) log a_n = a + (n − 1)d

ϕ(d) log(a + (n − 1)d) ∼ dn

ϕ(d) log n as n → ∞.

It follows that ω_a,d(x) = dx/(ϕ(d) log x) for x ∈ (1, +∞).

Example 2.4. Let (a_n) be a sequence defined as a_n= 2^pn−1, where p_nis the nth prime.

The numbers a_nare called Mersene numbers. A prime that appears in the sequence (a_n) is called Mersenne prime. Namely, it is easy to show (see, e.g., [36, p. 28]) that if 2ⁿ− 1 is prime, then so is n. The greatest known Mersenne prime is 2^43112609 − 1 with the exponent 43112609 (12978169 digit number), and it is discovered in August 2008. This is in fact one between 45 known Mersenne primes, and so a₄₅ ≤ 2^43112609− 1.

In 1980 H. Lenstra and C. Pomerance, working independently, came the conclusion that the probability that a Mersenne number 2^p− 1 is prime is e^γlog(ap)/(p log 2) with γ = 0.577216 . . . (the Euler-Mascheroni constant), where a = 2 if p ≡ 3(mod 4) and a = 6 if p ≡ 1(mod 4). Recall that the constant e^γ = 1.781072 . . . is important in num- ber theory; namely, e^γ = lim_{n→∞log p}¹

n

Qn k=1

pk

pk−1 which restates the third of Mertens’

theorems ([27], also see [23, pp. 351–353, Theorem 428]). Then notice that the distri- bution of the log of the Mersenne primes is a Poisson Process (see [45]).

Accordingly to the above assumption given by Lenstra and Pomerance, if ak = 2^qk−1, where (qk)^∞_k=1is a sequence of all primes ≡ 3( mod 4) (q1 = 3, q₂ = 7, q₃ = 11, . . .), for the associated function ω^(3,4)(x) to (a_k) we have that “the expected number” of primes between the first n terms of the sequence (q_k) is

(18) ∼ ω^(3,4)(n) ∼

n

X

k=1

e^γlog(2q_k)

qklog 2 as n → ∞.

Since q_k ∼ p_2k ∼ 2k log k, substituting this into (18) and using the well known asymp- totic formulaPn

k=11/k ∼ γ + log n as n → ∞, we get

ω^(3,4)(n) ∼ e^γ 2 log 2

n

X

k=2

log(4k log k)

k log k = e^γ 2 log 2

n

X

k=2

log k + log 4 + log log k k log k

∼ e^γ 2 log 2

n

X

k=2

1 k +

n

X

k=2

log 4 k log k +

n

X

k=2

log log k k log k

!

∼ e^γ 2 log 2



(γ + log n) + log 4 Z n

2

dx x log x+

Z n 2

log log x x log x dx



(the changes log x = s and log log x = t

= e^γ 2 log 2



log n + Z log n

log 2

ds s +

Z log log n log log 2

t dt



∼ e^γ 2 log 2



log n + log log n + (log log n)² 2



as n → ∞.

(19)

This shows that ω^(3,4)(x) = e^γ(log x + log log x + (log log x)²/2) /(2 log 2), and hence π_(ak)(a_n) ∼ e^γ/(2 log 2) (log n + log log n + (log log n)²/2). Substituting this in (4),

where (b_n) is the companion’ sequence of (a_k), and using the fact that q_n ∼ 2n log n, we find that

(20) b_n ∼ e^γn(log n)²

4^{n log n} as n → ∞.

Similarly, under the above assumptions attributed by Lenstra and Pomerance, if a⁰_k = 2^rk − 1, where (r_k)^∞_k=1 is a sequence of all primes ≡ 1(mod 4) (r₁ = 5, r₂ = 13, r₃ = 17, . . .), then for the associated function ω^(1,4)(x) to (a⁰_k) and the companion sequence (b⁰_n) of (a⁰_k) the same relations (18)–(20) are satisfied.

Example 2.5. Let (a_k)^∞_k=1be an increasing sequence of positive integers satisfying

(21) log a_k

a_k = o(k⁻¹).

Then from (4) and the obvious fact that π_(ak)(a_n) ≤ n for each n ∈N, we find that

(22) lim

n→∞b_n= 0.

In particular, (22) holds for any sequence (ak) satisfying one of the following asymp- totics: a_n ∼ aⁿ with a fixed a > 1; a_n ∼ n log^αn with α > 1; a_n ∼ n^α with α > 1; or a_n ∼ n^αlog^βn with α ≥ 1 and β > 1.

Accordingly, we ask the following question.

Question 2.6. For what real numbers α ∈ (0, 1) there exists a sequence (a_k) whose companion sequence(b_n) defined by (4) satisfies the limit relation

lim sup

n→∞

bn = α?

3. DISTRIBUTION OF PRIMES IN THE SEQUENCE(S_n)WITHS_n=P2n i=1p_i Here, as always in the sequel, we consider the distribution of primes in the sequence (S_n)^∞_n=1 whose terms are given by Sn =P2n

i=1p_i, where pi is the ith prime. Recall that the prime counting function π(x) is defined as the number of primes ≤ x.

Proposition 3.1. Let (S_n) be the sequence defined as S_n=P2n

i=1p_i. Then asn → ∞,

(23) S_n ∼ 2n²log n

and

(24) π(S_n) ∼ n².

Furthermore, ifx is a real number such that S_n≤ x < S_n+1, then

(25) n ∼

r x

log x as n → ∞.

Proof. Let (S_n⁰) be the sequence defined as S_n⁰ = Pn

i=1p_i (this is Sloane’s sequence A007504 in [42]). By the Prime Number Theorem, we have (see, e.g., [43, page 5]),

S_n⁰ :=

n

X

i=1

p_n ∼

n

X

k=1

k log k ∼ Z n

1

x log x dx = x² 2 log x

n 1

− Z n

1

x²

2 (log x)⁰dx

∼n²log n

2 as n → ∞.

(26)

It follows from (26) that

(27) S_n = S_2n⁰ ∼ 2n²log n,

which implies (23). By the Prime Number Theorem, from (27) we have (28) π(Sn) ∼ 2n²log n

log(2n²log n) ∼ 2n²log n

log 2 + 2 log n + log log n ∼ n².

Finally, (25) immediately follows from (23). 

Remark3.2. For refinements of the estimate (23), see [15], [39] and [41, Theorem 2.3]).

We see from (23) that there are ∼ n²primes less than Sn. Using this fact, Z.-W. Sun [43, Remark 1.6] conjectured that the number of primes in the interval (Pn

i=1p_i,Pn+1 i=1 p_i) is asymptotically equivalent to n/2 as n → ∞. Under the validity of this conjecture, in particular it follows that the number of primes in the interval (S_n, S_n+1) is asymptotically equivalent to n as n → ∞. Moreover, we also believe that the “probability” thatP2n

i=1p_i is a prime is 2n/p2n, which is ∼ 1/ log n because of p2n ∼ 2n log 2n. Notice that the

“probability” of a large integer n being a prime is also asymptotically equal to 1/ log n.

Furthermore, some computational results and heuristic arguments show that between these ∼ n² primes which are less than S_n there are ∼ 2n/ log S_n ∼ n/ log n primes that belong to the set S_n := {S₁, S₂, . . . , S_n}. For example, if n = 10⁸ then n/ log n = 10⁸/ log 10⁸ = 5428681.02, while from the second column of Table 1 of Section 6 we see that there are 5212720 primes in the set S₁₀⁸ (cf. Table 2 of Section 6). Accordingly, we propose the following curious conjecture which is basic in this paper.

Conjecture 3.3. The sequence (Sn) with Sn = P2n

i=1pi satisfies the Restricted Prime Number Theorem. In other words,

π_n : = π_(Sk)(S_n) = #{p : p is a prime and p = S_i for some i with 1 ≤ i ≤ n}

∼ n

log n as n → ∞.

(29)

Let us recall that in all results of this section (Corollaries 3.4, 3.6, 3.7, 3.8 and 3.13) we assume the truth of Conjecture 3.3. In particular, Conjecture 3.3 implies Euclid’s theorem (on the infinitude of primes) for (S_n) as follows.

Corollary 3.4 (Euclid’s theorem for the sequence (S_n)). The sequence (S_n) contains infinitely many primes.

Remark 3.5. Notice that the sequence (S_n) is closely related to the Sloane’s sequence A013918 [42] containing all primes (in increasing order) equal to the sum of the first m primes for some m ∈ N(A013918 is in fact the intersection of A000040-the sequence of all primes and A007504-sum of first n primes). The first few terms of the sequence A013918 are: 2, 5, 17, 41, 197, 281, 7699, 8893, 22039; see the related link by T. D. Noe [42, A013918] which gives the table of the first 10000 terms of this sequence (10000th term is 402638678093). Notice also that the Sloane’s sequence A013916 in [42] asso- ciated to the sequence A013918 gives numbers n such that the sum of the first n primes is prime. The first few terms of this sequence are: 1, 2, 4, 6, 12, 14, 60, 64, 96 (see the related link by D. W. Wilson [42, A013918] which gives table of the first 10000 terms of this sequence (10000th term is 244906). Similarly, the second Sloane’s sequence A013917 ((a_n)) associated to A013918, is defined as: a_nis prime and sum of all primes

≤ a_nis prime. The first few terms of this sequence are: 2, 3, 7, 13, 37, 43, 281.

As a further application of Conjecture 3.3, here we obtain the asymptotic expression for the kth prime in the sequence (S_n).

Corollary 3.6 (The asymptotic expression for the kth prime in the sequence (S_n)). Let q_k(k = 1, 2, . . .) be the kth prime in the sequence (S_n). Then

(30) q_k ∼ 2k²log³k as k → ∞.

Proof. If for a pair (k, n) there holds qk = S_n, then by Conjecture 3.3, we have

(31) k ∼ n

log n as n → ∞,

so that n ∼ k log n, and hence log n ∼ log k as n → ∞. Inserting this into (23), we find that

q_k = S_n ∼ 2n²log n ∼ 2(k log n)²log n = 2k²log³n ∼ 2k²log³k,

as desired. 

Corollary 3.7. Let q_k(k = 1, 2, . . .) be the kth prime in the sequence (S_n). Then

(32) qk∼ 2p²_klog k as k → ∞

and

(33) q_k∼ p_k²log²k as k → ∞.

Proof. From (30) and the fact that p_k∼ k log k we find that q_k∼ 2(k log k)²log k ∼ 2p²_klog k, which proves (32).

Similarly, from (30) and p_k² ∼ k²log k² = 2k²log k we find that q_k∼ (k²log k²) log²k ∼ p_k²log²k,

which implies (33). 

Furthermore, we have the following result.

Corollary 3.8. Let q_kbe thekth prime in the sequence (S_n) with q_k= S_n. Then

(34) lim

k→∞

k log k n = 1.

Proof. The asymptotic relation (31) implies that log n/ log k ∼ 1, which substituting in

(31) immediately gives (34). 

Motivated by some heuristic arguments and computations for some small integer val- ues d, we propose the following generalization of Conjecture 3.3.

Conjecture 3.9. For any fixed nonnegative integer d the sequence (Sn^(d))^∞_n=1defined as S_n^(d) = 2d + S_n = 2d +

2n

X

i=1

p_i, n = 1, 2, . . .

satisfies the Restricted Prime Number Theorem. In other words, asn → ∞, π_n^(d) :=π_(2d+Sk)(2d + S_n) = #{p : p is a prime and p = 2d + S_i

for some i with 1 ≤ i ≤ n} ∼ n log n. (35)

For d = 0 this conjecture is in fact Conjecture 3.3 (cf. Sloane’s sequence A013918 mentioned above).

Remark 3.10. Conjecture 3.3 and the fact that by (23) S_n ∼ 2n²log n imply that the average difference between consecutive primes in the sequence (S_n) near to 2n² is ap- proximately log(2n²) ∼ 2 log n.

Remark3.11. Numerous computational results concerning the sums of the first n primes (partial sums of consecutive primes) given by the Sloane’s sequence A007504 (here denoted as S_n⁰), and certain their curious arithmetical properties are presented in the following Sloane’s sequences in OEIS [42]: A051838, A116536, A067110, A067111, A045345, A114216, A024011, A077023, A033997, A071089, A083186, A166448, A196527, A065595, A165906, A061568, A066039, A077022, A110997, A112997, A156778, A167214, A038346, A038347, A054972, A072476, A076570, A076873, A077354, A110996, A123119, A189072, A196528, A022094, A024447, A121756, A143121, A117842, A118219, A131740, A143215, A161436, A161490, A013918 etc.

Since the sequence (Sn) is a subsequence of the sequence (S_n⁰) with S_n⁰ = Pn k=1pk

whose all terms with odd indices n are even integers, it follows that in accordance to Definition 1.1, Conjecture 3.3 is equivalent to

ω_(S⁰

k)(n) ∼ n 2 log n.

Therefore, Conjecture 3.3 is equivalent with the following one.

Conjecture 3.3’. Let (S_n⁰) be a sequence defined as S_n⁰ =Pn

k=1p_k,n = 1, 2, . . .. Then

(36) ω_(S⁰

k)(x) = x

2 log x f or x ∈ (1, ∞).

Proposition 3.12. For each n ≥ 3 we have

(37) 1 ≤ S_n

2n²log n < 1 + log 2

log n +log log(2n) log n .

Proof. By Mandl’s inequality (see, e.g., [39], [15]), for each n ≥ 9 there holds

(38) S_n⁰ < n

2p_n

(for a refinement of (38), see [17, the inequality 2.4]). Mandl’s inequality (38) with 2n instead of n becomes Sn < np_2n with n ≥ 5. This inequality together with the known inequality (see, e.g., [38, p. 69])

p_2n < 2n(log n + log 2 + log log(2n)) for all n ≥ 3 immediately yields

(39) S_n< 2n²(log n + log 2 + log log(2n)) for all n ≥ 5.

On the other hand, a lower bound for S_n⁰ can be obtained by using Robin’s inequality (see, e.g., [15, p. 51]) which asserts that for every n ≥ 2

(40) np_[n/2] ≤ S_n⁰.

The inequality (40) with 2n instead of n and the inequality n log n ≤ p_n with n ≥ 3 (see, e.g., [38, p. 69]) yield

(41) 2n²log n ≤ S_n for n ≥ 3.

The inequalities (39) and (41) immediately yield log n ≤ S_n

2n² < log n + log 2 + log log(2n) for all n ≥ 5, or equivalently,

(42) 1 ≤ S_n

2n²log n < 1 + log 2

log n +log log(2n)

log n for all n ≥ 5.

The inequalities given by (42) coincide with these of (37) for n ≥ 5. A direct calculation chows that (37) is also satisified for n = 3 and n = 4. This completes the proof.  Corollary 3.13. Let qk = Sm be thekth prime in the sequence (Sn)^∞_n=1. Then for all k ≥ 3 there holds

(43) 2m²log m < q_k < 2m²(log m + log(log(2m) + log 2).

Proof. The above inequalities coincide with (37) of Proposition 3.12 with n = m and

q_k = S_m. 

Remark3.14. Z.-W. Sun [44, the case α = 1 in Lemma 3.1] showed that for all n ≥ 2 S_n⁰ > 2 +n²log n

2



1 − 1

2 log n

 , which with 2n instead of n becomes

S_n > 2 + 2n²



log n + log 2

√e



≈ 2 + 2n²(log n + 0.193147), whence it follows that

S_n

2n²log n > 1 + 0.193147

n .

The above inequality is stronger that the left hand side of the inequality (37). Accord- ingly, if qk = S_m, then the first inequality of (43) can be refined in the form

q_k> 2 + 2m²(log m + 0.193147) for all k ≥ 3.

On the other hand, combining the inequalities (46) and (47) from the next section with the inequalities Sn > 2np_n and Sn < np_2n (given in proof of Proposition 3.12), respectively, we immediately obtain the following refinement of Proposition 3.12.

Proposition 3.15. For each n ≥ 3 there holds S_n

2n²log n ≥ 1 + log log n − 1

log n + log log n − 2.2 log²n , and for eachn ≥ 344192, we have

Sn

2n²log n ≤ 1 + log log(2n) + log 2 − 1

log n + log log(2n) − 2 (log n) log(2n).

Remark 3.16. If q_k = S_m, then in view of the first inequality of Proposition 3.15, the first inequality of (43) may be replaced by the following one:

q_k> 2m²



log m + log log m − 1 +log log m − 2.2 log m



for all k ≥ 3.

Remark3.17. The inequalities (38) and (40) and the asymptotic expression p_n∼ n log n show that the average of the first n primes is asymptotically equal to (n log n)/2 (cf.

Sloane’s sequence A060620 in [42]), that is, S_n⁰

n ∼ n log n

2 as n → ∞.

Conjecture 3.3 suggests the fact that for the sequence (Sn) would be valid the ana- logues of some other classical results and conjectures closely related to the Prime Num- ber Theorem and Riemann Hypothesis. In particular, if Q = {q1, q₂, . . . , q_k, . . .} is a set of all primes q₁ < q₂ < . . . < q_k < · · · in the sequence (S_n), it can be of interest to establish the asymptotic expression for q_kas k → ∞.

Finally, heuristic arguments, some computational results and Conjecture 3.3 lead to the follwing its generalization (cf. Sloane’s sequence A143121 - triangle read by rows, T (n, k) =Pn

j=kp_j, 1 ≤ k ≤ n; see the columns in Example of this sequence).

Conjecture 3.18. For any fixed positive integer k, let (Sn^(k)) := (Sn^(k))^∞_n=1 be the se- quence whosenth term is defined as

S_n^(k) =

2n+1

X

i=1

p_i+k, n ∈ N.

Then the sequence(Sn^(k)) satisfies the Restricted Prime Number Theorem.

For example, there are 78498 (resp. 664579) primes less than 10⁶ (resp. 10⁷), while the computations show that among the first 10⁶ (resp. 10⁷) terms of the sequences (Sn), (Sn^(k)) with k = 1, 2, . . . , 12 there are 69251 (resp. 594851), 69581 (resp. 594377), 68844 (resp. 593632), 68883 (resp. 595733), 69602 (resp. 596609), 69540 (resp.

596558), 69414 (resp. 595539), 69317 (resp. 594626), 69455 (resp. 595474), 69268 (resp. 594542), 68891 (resp. 593807), 69251 (resp. 594383), 69564 (resp. 595270) primes, respectively.

4. THE ASYMPTOTIC EXPRESSION FOR THEkTH PRIME IN THE SEQUENCE(S_n) As an easy consequence of the Prime Number Theorem, it can be deduced that p_n ∼ n log n as n → ∞ (see, e.g., [26]). Furthermore, a particular asymptotic expansion for p_n(see [26] or [34, the equality (66) of Section 6]; also see Sloane’s sequence A200265) yields

(44) pn = n



log n + log log n + O log log n log n



. It is also known that (see [16] and [38, p. 69])

(45) n(log n + log log n − 1) < p_n< n(log n + log log n).

A more precise work about this can be found in [37] and [40] where related results are as follows:

(46) n



log n + log log n − 1 + log log n − 2.2 log n



≤ pn for n ≥ 3 and

(47) pn ≤ n



log n + log log n − 1 + log log n − 2 log n



for n ≥ 688383.

The inequalities (46) and (47) immediately yield the following result.

Corollary 4.1. For each n ≥ 688383 the interval (48)

 n



log n + log log n − 1 + log log n − 2.2 log n

 , n



log n + log log n − 1 +log log n − 2 log n



contains at least one prime. Furthermore, the lenghtlnof this interval is

(49) ln= 0.2n

log n ∼ 0.2p_n log²n.

As an application of Conjecture 3.3, we obtain the following result.

Corollary 4.2. Let q_k be the kth prime in the sequence (S_n) with S_n = P2n

k=1p_k. If q_k = S_m, then under Conjecture3.3 there holds

(50) q_k ∼ 2k²log³k ∼ 2m²log m ∼ p_bk2log²kc as k → ∞, and

(51) lim

k→∞

k log k m = 1.

Proof. The first asymptotic relation of (50) coincides with (30) of Corollary 3.6. Further, by (37) of Proposition 3.12, we have

(52) q_k= S_m ∼ 2m²log m.

Moreover, we have

(53) p_[k2log²k] ∼ k²(log²k) log(k²log²k) ∼ 2k²log³k as k → ∞.

The last two asymptotic expressions of (50) follow from (52) and (53).

It remains to prove (51). If we suppose that (51) is not satisfied, then there exists ε > 0 and an infinite subsequence (k_j, m_j)^∞_j=1 of the sequence (k, m)^∞_k=1such that k_jlog k_j ≥ (1 + ε)m_j for all j ∈Nor kjlog k_j ≤ (1 − ε)m_j for all j ∈ N. In the first case, using (50) for all sufficiently large j, we find that

m²_jlog m_j ∼ k²_j log³k_j ≥ (1 + ε)²m²_jlog k_j,

whence we immediately get log m_j ≥ (1 + ε)²log k_j, or equivalently, m_j ≥ k_j^t with t = (1 + ε)² > 1. From the previous inequality and the fact that t > 2 we have

m²_j log m_j > m²_j ≥ k_j^2t  k²_jlog³k_j,

which contradicts the fact that by (50) 2k²log³k ∼ 2m²log m. In a similar way as in the first case, in the second case we find that m_j ≤ k^s_j for a constant s = (1 − ε)² < 1.

Then choosing a sufficiently large j₀ such that log m_j < m^1/s−1_j for all j > j₀, in view of the fact that s + 1 < 2 we get

m²_jlog mj < m^1+1/s_j ≤ k_j^s+1 k_j²log³kj.

A contradiction, and therefore, (51) is true. 

Remark4.3. From (50) and p_k ∼ k log k we see that

(54) q_k

2k log²k ∼ k log k ∼ p_k as k → ∞.

The above asymptotic expression together with the assumption that Conjecture 3.3 is true suggests the fact that for the sequence qk/(k log²k
) would be satisfied the asymptotic expansion similar to (44), i.e.,

(55) q_k

2k log²k = k(log k + log log k + Q_k),

with some sequence (Q_k). Motivated by (55), we establish the following asymptotic expression for the kth prime in the sequence (S_n).

Theorem 4.4 (The asymptotic expression for the kth prime in the sequence (S_n)). Let q_k be thekth prime in the sequence (S_n) (k = 2, 3, . . .). Then under Conjecture 3.3 there exists a sequence(M_k) of positive rael numbers such that lim_k→∞M_k = 1 and

(56) q_k = 2M_k⁵k²log²k(log k + log log k + 2 log M_k).

For the proof of Theorem 4.4 we will need the following result.

Lemma 4.5. Let S_m = q_kbe thekth prime in the sequence (S_n). Then under Conjecture (3, 3) we have

(57) q_k ∼ 2m²√

m log m

√k log k as k → ∞.

Proof. First notice that, under notations of Lemma 4.5, Conjecture 3.3 yields (cf. (51) of Corollary 4.2)

(58) k ∼ m

log m as k → ∞.

Using (58), we find that (59) 2m²√

m log m

√k log k ∼ 2m²√

m log m r

m

1 −^{log log m}_{log m} 

∼ 2m²log m as k → ∞.

The asymptotic relation (59) and the fact that by (50) of Corollary 4.2, qk = S_m ∼

2m²log m immediately yield (57). 

Proof of Theorem4.4. Let (C_k)^∞_k=2 be a sequence of positive real numbers such that m(k) = m = C_kk log k with k ≥ 2 and q_k = S_m. Then by (51) of Corollary 4.2, we have C_k → 1 as k → ∞. Taking m = C_kk log k into (57) of Lemma 4.5, as k → ∞ we obtain that

q_k∼ 2 q

C_k⁵k⁵log⁵k(log k + log log k + log Ck)

√k log k

= 2C_k²p

C_kk²log²k(log k + log log k + log C_k) =: f (k, C_k).

(60)

Let (δk) be a positive real sequence such that

(61) q_k = δ_kf (k, C_k) for each k ≥ 2.

Then from (60) we see that δ_k → 1 as k → ∞. For a fixed k ≥ 2 consider the equation f_k(x) = δ_kf (k, C_k) which can be written in the form

(62) x²√

x(log k + log log k + log x) = δ_kC_k²p

C_k(log k + log log k + log C_k).

Notice that for any fixed integer k ≥ 2, the real function fk(x) defined as f_k(x) = 2x²√

x(log k + log log k + log x), x > 0,

satisfies the limit relations limx→+∞fk(x) = +∞ and limx→+0fk(x) = 0. From this it can be easily shown that for each integer k ≥ 2 the equation (62) has a positive real solution xk. Using the facts that limx→+∞C_k = lim_x→+∞δ_k=1, it can be easily show that limk→∞x_k = 1. Then taking x_k = M_k²(k = 2, 3, . . .), then limk→∞M_k = 1 and by (62) we find that fk(M_k²) = δ_kf (k, C_k) = q_k, whence it follows that

q_k = 2M_k⁵k²log²k(log k + log log k + 2 log M_k).

This proves (56) and the proof is completed. 

Computational results (cf. the eighth column of Table 1 of Section 6) suggest the additional relationship between k’s and m’s as follows.

Conjecture 4.6. For each pair (k, m) with k ≥ 1 and q_k = S_mwe have

(63) bk log kc + 1 ≤ m,

or equivalently,

(64) qk≥ Sbk log kc+1.

Furthermore, for eachk ≥ 10⁴,

(65) m ≤ b1.4k log kc,

or equivalently,

(66) qk≤ Sb1.4k log kc.

Corollary 4.7. If the inequality (63) of Conjecture 4.6 is true, then for each k ≥ 1 there holds

(67) q_k > 2k²(log²k)(log k + log log k).

Proof. Combining the inequality (63) with the inequality on the left hand side of (37) of Proposition 3.12, we find that

qk = Sm ≥ Sbk log kc+1 ≥ 2(bk log kc + 1)²log(bk log kc + 1)

> 2k²(log²k) log(k log k) = 2k²(log²k)(log k + log log k), (68)

as desired. 

Corollary 4.8. If the inequality (63) of Conjecture 4.6 is true, then M_k > 1 for each k ≥ 1, where (M_k) is the sequence defined by (56) of Theorem 4.4.

Proof. The assertion follows immediately from the inequality (67) and the expression

(56) for q_kgiven by Theorem 4.4. 

Finally, in view of the data of the last column in Table 1 of Section 6 and some con- siderations presented above, we propose the following conjecture which is stronger than Corollary 4.7.

Conjecture 4.9. For every k ≥ 252028 with q_k = S_mthere holds

(69) q_k > 2m²√

m log m

√k log k .

In view of the well known inequality p_k > k(log k + log log k), the following conjec- ture is also stronger than Corollary 4.7.

Conjecture 4.10. There exists k₀ ∈_Nsuch that for everyk ≥ k₀ there holds q_k > 2kp_klog²k,

wherep_kis thekth prime.

Remark4.11. The last column of Table 1 presented in Section 6 shows that q_k ≈ 2m²√

m log m

√k log k

is a “good” approximation for the kth prime sum q_k. Notice that this approximation can be written as

qk≈ 2m²log m ·

r m

k log k,

where the valuespm/(k log k) slowly tend to 1 as k grows. In particular, from the last row of Table 1 of Section 6 we see that for m = 10⁹− 2 (i.e., for k = 46388006) we have pm/(k log k) ≈ 1.105079. Hence, in view of the above approximation, we believe that for all values m up to 10⁹ there holds

q_k > 2.2m²log m.

Notice that some values of the sequences (Q⁰_k) such that

Q⁰_k = q_k− 2k²(log²k)(log k + log log k) 2k²log²k log log k

and the sequence (Q⁰⁰_k) such that

Q⁰⁰_k= q_k− 2(p_k)²log k 2k²log²k log log k

are given in Table 3 of Section 6. Table 3 also shows that for almost all values m up to 10⁹(i.e., for k ≤ 46388006) there holds

kp_k> 1.1m²log m.

Finally, Table 2 of Section 6 leads to the following conjecture whose both parts are obviously stronger than Conjecture 4.6.

Conjecture 4.12. Let π(x) be the prime counting function, and let πnbe the number of primes in the set{S₁, S₂, . . . , S_n}. Then

π_n < π(n) for for eachn ≥ 10⁴ and

π_n< n log n for eachn ≥ 10⁵.

5. ESTIMATIONS OF VALUES M_kFROMTHEOREM4.4

The computational results related to the search of primes in the sequence (Sn) given in the following section (Table 1) and some heuristic arguments suggest the fact that the sequence (Sn/(2n²log n))^∞_n=2 plays an important role for estimating the values Mk

(k = 1, 2, . . .) in the expression (56) for the kth prime q_kin the sequence (S_n).

Here we first consider the sequence (S_n/(2n²)).

Proposition 5.1. The sequence (vn) defined as

(70) v_n = S_n

2n², n ∈N, is increasing forn ≥ 2.

Proof. Since Sn+1 = S_n+ p_2n+1 + p_2n+2, an easy calculation shows that rn < r_n+1 is equivalent with

S_n

2n² < p_2n+1+ p_2n+2 2(2n + 1) , which can be written as

(71) p_2n+1+ p_2n+2

2 > S_n 1 n + 1

2n²

 .

By a refinement of Mandl’s inequality due to Hassani [17], for every n ≥ 10 we have

(72) n

2p_n−

n

X

i=1

p_i > 0.01659n². Replacing n by 2n into (72) it becomes

(73) p_2n− S_n

n > 0.06636n² for all n ≥ 5.

Further, by the inequality (37) of Proposition 3.12 we have (74) log(2n) + log log(2n) > Sn

2n² for all n ≥ 5.

By using Mathematica 8, it is easy to to prove the inequality (75) 0.06636n² > log(2n) + log log(2n) for all n ≥ 8.

Finally, combining the inequalities (73), (74), (75) and the obvious inequality (p2n+1+ p_2n+2)/2 > p_2n immediately gives (71) for all n ≥ 8. This together with a direct verification that vn< v_n+1for 2 ≤ n ≤ 8 concludes the proof.  Remark 5.2. Notice that the sequence (v_n) defined by (70) is a subsequence of the se- quence (v⁰_n) defined as

v_n⁰ = 2S_n⁰

n² := 2Pn i=1p_i

n² , n ∈ N;

namely, vn = v⁰_2nfor all n = 1, 2, . . .. Similarly as in the proof of Proposition 5.1, it can be shown that the sequence (v⁰_n) is increasing for n ≥ 4.

Contrary to Proposition 5.1, we propose the following conjecture.